Spreads Being Altered When Adding Pages

I have a few booklets created in InDesign.  The area standard booklets so they have two-page spreads for facing pages on all but the first and last pages.

Whenever I add pages from one booklet to another booklet (for example, to make a file which has multiple booklets in one), it wreaks havoc with my spreads.  Suddenly I find that some spreads are missing a page. And InDesign won't let me fix it when I try to drag the page back into the spread or even add a page to replace the missing page.

Does anyone know why InDesign is doing this and how to stop it from doing this?